The time dimensional reduction method to determine the initial conditions without the knowledge of damping coefficients

TL;DR

This paper introduces a novel method for reconstructing initial conditions of hyperbolic equations with unknown damping by reducing the problem to elliptic equations using Fourier expansion and solving with Carleman contraction, demonstrated through numerical examples.

Contribution

The paper presents a new time dimensional reduction technique that reconstructs initial conditions without prior damping coefficient knowledge, improving computational efficiency and accuracy.

Findings

Accurate initial condition reconstruction demonstrated.

Method effectively eliminates the need for damping coefficient knowledge.

Numerical examples confirm computational efficiency and precision.

Abstract

This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coefficient. Our approach involves approximating the hyperbolic equation's solution by its truncated Fourier expansion in the time domain and using a polynomial-exponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The method not only delivers precise solutions but also showcases remarkable computational efficiency.